Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\cot ^{2} t}{\csc t} d t $$

Short answer

The integral \(\int \frac{\cot ^{2} t}{\csc t} dt\) equals to \(-\frac{1}{2sin^2t} + C\).

Step-by-step solution

Simplify the Integral Expression

The first step in solving an integral problem is simplification. In this case, use the basic definition of cotangent and cosecant to rewrite the fraction. \(cot^2t\) is the same as \(\frac{cos^2t}{sin^2t}\) and \(csc t\) is the same as \(\frac{1}{sin t}\). Putting it all together we get \(\frac{cos^2t}{sin^3t}\).

Integration Trigonometric Function

Let \(sin t = u\). Therefore, \(cos t dt = du\), and the integral \( \frac{cos^2t}{sin^3t} dt\) becomes \( \int \frac{1}{u^3} du\). The integral of \(\frac{1}{u^3}\) with respect to u can be directly found using the power rule for integration, which results in \(-\frac{1}{2u^2}\). The constant of integration C is appended at the end.

Substituting Back the Value of 'u'

To return to the original variable of the problem 't', we substitute back \(sin t\) for 'u'. So the final result is \(-\frac{1}{2sin^2t} + C\)

Confirming the Result Using a Computer Algebra System

You can use a computer algebra system like Mathematica or sympy in Python to check the above result. Simply plug in the original expression and see if it returns \(-\frac{1}{2sin^2t} + C\) as the answer.